A134058 -id:A134058 - cp's OEIS frontend

A134060 Triangle T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k), read by rows.

1, 2, 3, 2, 6, 3, 2, 9, 9, 3, 2, 12, 18, 12, 3, 2, 15, 30, 30, 15, 3, 2, 18, 45, 60, 45, 18, 3, 2, 21, 63, 105, 105, 63, 21, 3, 2, 24, 84, 168, 210, 168, 84, 24, 3, 2, 27, 108, 252, 378, 378, 252, 108, 27, 3

Offset: 0

Gary W. Adamson, Oct 05 2007

			First few rows of the triangle are:
  1;
  2,  3;
  2,  6,  3;
  2,  9,  9,  3;
  2, 12, 18, 12,  3;
  2, 15, 30, 30, 15, 3;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A052940 (row sums), A127927, A134058.

[1] cat [k eq 0 select 2 else 3*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[3*Binomial[n, k] -Boole[k==0] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

def A134060(n,k): return 3*binomial(n,k) -bool(k==0) -bool(n==0)
flatten([[A134060(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k) as infinite lower triangular matrices.
Sum_{k=0..n} T(n, k) = A052940(n).
T(n, k) = 3*binomial(n,k) - [k=0] - [n=0]. - G. C. Greubel, May 03 2021

A135087 Triangle T(n, k) = 2*A134058(n, k) - 1, read by rows.

Original entry on oeis.org

1, 3, 3, 3, 7, 3, 3, 11, 11, 3, 3, 15, 23, 15, 3, 3, 19, 39, 39, 19, 3, 3, 23, 59, 79, 59, 23, 3, 3, 27, 83, 139, 139, 83, 27, 3, 3, 31, 111, 223, 279, 223, 111, 31, 3, 3, 35, 143, 335, 503, 503, 335, 143, 35, 3

Offset: 0

Gary W. Adamson, Nov 18 2007

			First few rows of the triangle are:
  1;
  3,  3;
  3,  7,  3;
  3, 11, 11,  3;
  3, 15, 23, 15,  3;
  3, 19, 39, 39, 19,  3;
  3, 23, 59, 79, 59, 23, 3;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A095768, A134058.

[1] cat [4*Binomial(n,k) -1: k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[4*Binomial[n, k] -2*Boole[n==0] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

def A135087(n,k): return 4*binomial(n,k) -2*bool(n==0) -1
flatten([[A135087(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n, k) = 2*A134058(n, k) - 1.
From G. C. Greubel, May 03 2021: (Start)
T(n, k) = 4*binomial(n, k) - 2*[n=0] - 1.
Sum_{k=0..n} T(n, k) = 2^(n+2) - (n + 1 + 2*[n=0]) = A095768(n) - 2*[n=0]. (End)

A046055 Orders of finite Abelian groups having the incrementally largest numbers of nonisomorphic forms (A046054).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 221184, 262144, 442368, 524288, 663552, 884736, 995328, 1048576, 1327104, 1769472, 1990656, 2097152, 2654208, 3538944, 3981312, 4194304

Offset: 1

Eric W. Weisstein

Different from A151821, but often confused with it.

Nicolas used the notation a(n) for the number of Abelian groups of order n (A000688) and named these numbers a-highly composite numbers (a-hautement composés). - Amiram Eldar, Aug 20 2019

Amiram Eldar, Table of n, a(n) for n = 1..1111 (terms 1..216 from Charlie Neder)
H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Jean-Louis Nicolas, Sur les entiers N pour lesquels il y a beaucoup de groupes abéliens d’ordre N, Annales de l'institut Fourier. Vol. 28, No. 4. (1978), pp. 1-16, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.
Wikipedia, Abelian group
Index entries for sequences related to groups

Cf. A000079, A000688, A046054, A046056, A010684.

Warning: this is different from A151821.

aa = {}; max = 0; Do[If[FiniteAbelianGroupCount[n] > max, max = FiniteAbelianGroupCount[n]; AppendTo[aa, n]], {n, 2^22}]; aa (* Artur Jasinski, Oct 06 2011 *)

Warning: the g.f. is not x*(1+2*x)/(1-2*x), as claimed earlier.
Warning: this is not the binomial transform of A010684, as claimed earlier.
Warning: this is not the row sums of either A131127 or A134058, as claimed earlier.

More terms from David Wasserman, Feb 06 2002
Many incorrect formulas and assertions deleted by R. J. Mathar, Jul 08 2009
Edited by N. J. A. Sloane, Jul 08 2009

A028326 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2, 2, 20, 90, 240, 420, 504, 420, 240, 90, 20, 2, 2, 22, 110, 330, 660, 924, 924, 660, 330, 110, 22, 2

Offset: 0

Mohammad K. Azarian

Also number of binary vectors of length n+1 with k+1 runs (1 <= k <= n).
If the last two entries in each row are removed and 0 replaces the entries in a checkerboard pattern, we obtain
  2;
  0,  6;
  2,  0, 12;
  0, 10,  0,  20;
  2,  0, 30,   0,  30;
  0, 14,  0,  70,   0,  42;
  2,  0, 56,   0, 140,   0, 56;
  0, 18,  0, 168,   0, 252,  0, 72;
  ...
This plays the same role of recurrence coefficients for second differences of polynomials as triangle A074909 plays for the first differences. - R. J. Mathar, Jul 03 2013
From Roger Ford, Jul 06 2023: (Start)
T(n,k) = the number of closed meanders with n top arches, n+1 exterior arches and with k = the number of arches of length 1 - (n+1).
Example of closed meanders with 4 top arches and 5 exterior arches:
   exterior arches are top arches or bottom arches without a covering arch
   /\ = top arch length 1, \/ = bottom arch length 1
                                    
     /  \       Top: /\=3      /  \    /  \   Top: /\=2
/\  / /\ \  /\                / /\ \  / /\ \
\ \/ /  \ \/ /  Bottom: \/=2  \/  \ \/ /  \/  Bottom: /\=3
 \/    \/   k=5-5=0            \/       k=5-5=0          T(4,0) = 2
      ____                    
     /      \   Top: /\=3      /  \           Top: /\=3
/\  / /\  /\ \                / /\ \  /\  /\
\ \/ /  \/  \/  Bottom: \/=3  \/  \ \/  \/ /  Bottom: \/=3
 \/           k=6-5=1            \____/   k=6-5=1
  ____                                
 /      \       Top: /\=3              /  \   Top: /\=3
/ /\  /\ \  /\                /\  /\  / /\ \
\/  \/  \ \/ /  Bottom: \/=3  \ \/  \/ /  \/  Bottom: \/=3
         \/   k=6-5=1        \____/       k=6-5=1         T(4,1) = 4
  ________
 /          \   Top: /\=3
/ /\  /\  /\ \                 /\  /\  /\  /\ Top: /\=4
\/  \/  \/  \/  Bottom: \/=4   \ \/  \/  \/ / Bottom: ||=3
                k=7-5=2         \________/  k=7-5=2         T(4,2) = 2.
(End)

			Triangle begins:
  2;
  2,  2;
  2,  4,   2;
  2,  6,   6,   2;
  2,  8,  12,   8,   2;
  2, 10,  20,  20,  10,    2;
  2, 12,  30,  40,  30,   12,    2;
  2, 14,  42,  70,  70,   42,   14,    2;
  2, 16,  56, 112, 140,  112,   56,   16,   2;
  2, 18,  72, 168, 252,  252,  168,   72,  18,   2;
  2, 20,  90, 240, 420,  504,  420,  240,  90,  20,   2;
  2, 22, 110, 330, 660,  924,  924,  660, 330, 110,  22,  2;
  2, 24, 132, 440, 990, 1584, 1848, 1584, 990, 440, 132, 24, 2;

I. Goulden and D. Jackson, Combinatorial Enumeration, John Wiley and Sons, 1983, page 76.

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 48.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A028327, A028328, A028329, A028330, A028331, A028332, A124927, A134058.

a028326 n k = a028326_tabl !! n !! k
a028326_row n = a028326_tabl !! n
a028326_tabl = iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [2]
-- Reinhard Zumkeller, Mar 12 2012

[2*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 27 2021

T := proc(n, k) if k=0 then 2 elif k>n then 0 else T(n-1, k)+T(n-1, k-1) fi end:
for n from 0 to 13 do seq(T(n, k), k=0..n) od; # Zerinvary Lajos, Dec 16 2006

Table[2*Binomial[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Robert G. Wilson v, Mar 05 2012 *)

T(n,k) = 2*binomial(n,k) \\ Charles R Greathouse IV, Feb 07 2017

from sympy import binomial
def T(n, k):
    return 2*binomial(n, k)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 29 2017

flatten([[2*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021

G.f. for the number of length n binary words with k runs: (1-x+x*y)/(1-x-x*y) [Goulden and Jackson]. - Geoffrey Critzer, Mar 04 2012

More terms from Donald Manchester, Jr. (s1199170(AT)cedarnet.cedarville.edu)

A134059 Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 3, 3, 3, 6, 3, 3, 9, 9, 3, 3, 12, 18, 12, 3, 3, 15, 30, 30, 15, 3, 3, 18, 45, 60, 45, 18, 3, 3, 21, 63, 105, 105, 63, 21, 3, 3, 24, 84, 168, 210, 168, 84, 24, 3, 3, 27, 108, 252, 378, 378, 252, 108, 27, 3

Offset: 0

Gary W. Adamson, Oct 05 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [3, -2, 0, 0, 0, 0, 0, ...] DELTA [3, -2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007

			First few rows of the triangle:
  1;
  3,  3;
  3,  6,  3;
  3,  9,  9,  3;
  3, 12, 18, 12,  3;
  3, 15, 30, 30, 15,  3;
  3, 18, 45, 60, 45, 18,  3;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5150 (rows n = 0..100, flattened)

Cf. A007318, A082505 (row sums), A084938, A134058.

A134059:= func< n,k | n eq 0 select 1 else 3*Binomial(n,k) >;
[A134059(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021

Join[{1},Rest[Flatten[Table[3Binomial[n,k],{n,0,10},{k,0,n}]]]] (* Harvey P. Dale, Feb 15 2014 *)
Table[3*Binomial[n,k] -2*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)

def A134059(n,k): return 3*binomial(n,k) - 2*bool(n==0)
flatten([[A134059(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021

3*Pascal's triangle A007318, then replace T(0,0) with 1.
G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k) *x^n * y^k = 1 - 3*(1+y)*x/(x+x*y-1). - R. J. Mathar, Feb 19 2020
From G. C. Greubel, Apr 27 2021: (Start)
T(n, k) = 3*binomial(n,k) - 2*[n=0].
Sum_{k=0..n} T(n, k) = 3*2^n - 2*[n=0] = A082505(n+1). (End)
E.g.f.: 3*exp(x*(1+y)) - 2. - Stefano Spezia, Apr 03 2024

Title changed by G. C. Greubel, Apr 26 2021

A132200 Numbers in (4,4)-Pascal triangle .

Original entry on oeis.org

1, 4, 4, 4, 8, 4, 4, 12, 12, 4, 4, 16, 24, 16, 4, 4, 20, 40, 40, 20, 4, 4, 24, 60, 80, 60, 24, 4, 4, 28, 84, 140, 140, 84, 28, 4, 4, 32, 112, 224, 280, 224, 112, 32, 4, 4, 36, 144, 336, 504, 504, 336, 144, 36, 4, 4, 40, 180, 480, 840, 1008, 840, 480, 180, 40, 4

Offset: 0

Philippe Deléham, Nov 19 2007

This triangle belongs to the family of (x,y)-Pascal triangles ; other triangles arise by choosing different values for (x,y): (1,1) -> A007318 ; (1,0) -> A071919 ; (3,2) -> A029618 ; (2,2) -> A134058 ; (-1,1) -> A112467 ; (0,1) -> A097805 ; (5,5) -> A135089 ; etc..

			Triangle begins:
  1;
  4,  4;
  4,  8,  4;
  4, 12, 12,  4;
  4, 16, 24, 16,  4;
  4, 20, 40, 40, 20, 4;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A007318, A134058, A134059, A135089.

[1] cat [4*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[4*Binomial[n,k] -3*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

def A132200(n,k): return 4*binomial(n,k) - 3*bool(n==0)
flatten([[A132200(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n,k) = 4*binomial(n,k), n>0 ; T(0,0)=1.

Sum_{k=0..n} T(n,k) = 2^(n+2) - 3*[n=0]. - G. C. Greubel, May 03 2021

A136521 Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Gary W. Adamson, Jan 02 2008

			First few rows of the triangle are:
  1;
  0, 2;
  0, 0, 2;
  0, 0, 0, 2;
  0, 0, 0, 0, 2;
  ...

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A001263, A007318, A040000, A124927, A134058, A136522.

Table[2*Boole[k==n] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

flatten([[2*bool(k==n) -bool(n==0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

By columns, (1, 0, 0, 0, ...) in leftmost column; all others are (2, 0, 0, 0, ...).
By rows, row 1 = 1, others = (n-1) zeros followed by "2".
A007318(n,k) * T(n,k) = A124927(n,k).
T(n,k) * A007318(n,k) = A134058(n,k).
A001263(n,k) * T(n,k) = A136522(n,k).
From G. C. Greubel, May 03 2021: (Start)
T(n, k) = 2*[k=n] - [n=0].
Sum_{k=0..n} T(n, k) = A040000(n). (End)

More terms added by G. C. Greubel, May 03 2021

A173048 Triangle T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 2, read by rows.

Original entry on oeis.org

1, 3, 3, 9, 12, 9, 65, 70, 70, 65, 1025, 990, 560, 990, 1025, 32769, 31806, 11160, 11160, 31806, 32769, 2097153, 2064510, 671832, 178560, 671832, 2064510, 2097153, 268435457, 266338558, 87413592, 12850368, 12850368, 87413592, 266338558, 268435457

Offset: 0

Roger L. Bagula, Feb 08 2010

			Triangle begins as:
        1;
        3,       3;
        9,      12,      9;
       65,      70,     70,     65;
     1025,     990,    560,    990,   1025;
    32769,   31806,  11160,  11160,  31806,   32769;
  2097153, 2064510, 671832, 178560, 671832, 2064510, 2097153;

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A134058 (q=1), this sequence (q=2), A173049 (q=3).

R:=PowerSeriesRing(Integers(), 50);
p:= func< x,n,q | n eq 0 select 1 else (&*[x+q^j: j in [1..n]]) + (&*[1+q^j*x: j in [1..n]]) >;
T:= func< n,q | Coefficients(R!( p(x,n,q) )) >;
[T(n,2): n in [0..10]]; // G. C. Greubel, Apr 26 2021

p[x_, n_, q_]:= If[n==0, 1, Product[x+q^j, {j,n}] + Product[x*q^j +1, {j,n}]];
T[n_, k_, q_]:= SeriesCoefficient[p[x,n,q], {x,0,k}];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 26 2021 *)

T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 2.

Edited by G. C. Greubel, Apr 26 2021

A173049 Triangle T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 3, read by rows.

Original entry on oeis.org

1, 4, 4, 28, 24, 28, 730, 390, 390, 730, 59050, 29280, 7020, 29280, 59050, 14348908, 7145292, 914760, 914760, 7145292, 14348908, 10460353204, 5223003240, 650485836, 49397040, 650485836, 5223003240, 10460353204, 22876792454962, 11433166054158, 1427188022442, 55340738838, 55340738838, 1427188022442, 11433166054158, 22876792454962

Offset: 0

Roger L. Bagula, Feb 08 2010

			Triangle begins as:
            1;
            4,          4;
           28,         24,        28;
          730,        390,       390,      730;
        59050,      29280,      7020,    29280,     59050;
     14348908,    7145292,    914760,   914760,   7145292,   14348908;
  10460353204, 5223003240, 650485836, 49397040, 650485836, 5223003240, 10460353204;

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A134058 (q=1), A173048 (q=2), this sequence (q=3).

R:=PowerSeriesRing(Integers(), 50);
p:= func< x,n,q | n eq 0 select 1 else (&*[x+q^j: j in [1..n]]) + (&*[1+q^j*x: j in [1..n]]) >;
T:= func< n,q | Coefficients(R!( p(x,n,q) )) >;
[T(n,3): n in [0..10]]; // G. C. Greubel, Apr 26 2021

p[x_, n_, q_]:= If[n==0, 1, Product[x+q^j, {j,n}] + Product[x*q^j +1, {j,n}]];
T[n_, k_, q_]:= SeriesCoefficient[p[x,n,q], {x,0,k}];
Table[T[n, k, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 26 2021 *)

T(n, k) = [x^k](p(x, n, q)) where p(x,n,q) = Product_{j=1..n} (x + q^j) + Product_{j=1..n} (x*q^j + 1), p(x, 0, q) = 1, and q = 3.

Edited by G. C. Greubel, Apr 26 2021

A135089 Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.

Original entry on oeis.org

1, 5, 5, 5, 10, 5, 5, 15, 15, 5, 5, 20, 30, 20, 5, 5, 25, 50, 50, 25, 5, 5, 30, 75, 100, 75, 30, 5, 5, 35, 105, 175, 175, 105, 35, 5, 5, 40, 140, 280, 350, 280, 140, 40, 5, 5, 45, 180, 420, 630, 630, 420, 180, 45, 5, 5, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 5

Offset: 0

Gary W. Adamson, Nov 18 2007

Row sums = A020714 (except for the first term).

Triangle T(n,k), 0 <= k <= n, read by rows given by (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 24 2013

			First few rows of the triangle:
  1;
  5,  5;
  5, 10,  5;
  5, 15, 15,   5;
  5, 20, 30   20,  5;
  5, 25, 50,  50, 25,  5;
  5, 30, 75, 100, 75, 30, 5.

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A007318, A020714, A132200, A134058, A134059.

[1] cat [5*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[5*Binomial[n,k] -4*Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 22 2016; May 03 2021 *)

def A135089(n,k): return 5*binomial(n,k) - 4*bool(n==0)
flatten([[A135089(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n,k) = 5*binomial(n,k), n > 0, 0 <= k <= n.
Equals 2*A134059(n,k) - A007318(n,k).
G.f.: (1+4*x+4*x*y)/(1-x-x*y). - Philippe Deléham, Nov 24 2013
Sum_{k=0..n} T(n,k) = A020714(n) - 4*[n=0]. - G. C. Greubel, May 03 2021

cp's OEIS Frontend

A134060 Triangle T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k), read by rows.

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A135087 Triangle T(n, k) = 2*A134058(n, k) - 1, read by rows.

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A046055 Orders of finite Abelian groups having the incrementally largest numbers of nonisomorphic forms (A046054).

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A028326 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).

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A134059 Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.

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A132200 Numbers in (4,4)-Pascal triangle .

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